A Sobolev inequality and the individual invariance principle for diffusions in a periodic potential

We consider a diffusion process in $\mathbb{R}^d$ with a generator of the form $L:=\frac 12 e^{V(x)}div(e^{-V(x)}\nabla )$ where $V$ is measurable and periodic. We only assume that $e^V$ and $e^{-V}$ are locally integrable. We then show that, after proper rescaling, the law of the diffusion converges to a Brownian motion for Lebesgue almost all starting points. This pointwise invariance principle was previously known under uniform ellipticity conditions (when $V$ is bounded), and was recently proved under more restrictive $L^p$ conditions on $e^V$ and $e^{-V}$. Our approach uses Dirichlet form theory to define the process, martingales and time changes and the construction of a corrector. Our main technical tool to show the sub-linear growth of the corrector is a new weighted Sobolev type inequality for integrable potentials. We heavily rely on harmonic analysis technics.